Global optimization of univariate Lipschitz functions: II. New algorithms and computational comparison

We consider the following global optimization problems for a Lipschitz functionf implicitly defined on an interval [a, b]. Problem P: find a globally-optimal value off and a corresponding point; Problem Q: find a set of disjoint subintervals of [a, b] containing only points with a globally-optimal value and the union of which contains all globally optimal points. A two-phase algorithm is proposed for Problem P. In phase I, this algorithm obtains rapidly a solution which is often globally-optimal. Moreover, a sufficient condition onf for this to be the case is given. In phase II, the algorithm proves the-optimality of the solution obtained in phase I or finds a sequence of points of increasing value containing one with a globally-optimal value. The new algorithm is empirically compared (on twenty problems from the literature) with a best possible algorithm (for which the optimal value is assumed to be known), with a passive algorithm and with the algorithms of Evtushenko, Galperin, Shen and Zhu, Piyavskii, Timonov and Schoen. For small, the new algorithm requires only a few percent more function evaluations than the best possible one. An extended version of Piyavskii's algorithm is proposed for problem Q. A sufficient condition onf is given for the globally optimal points to be in one-to-one correspondance with the obtained intervals. This result is achieved for all twenty test problems.The research of the authors has been supported by AFOSR grants 0271 and 0066 to Rutgers University. Research of the second author has been also supported by NSERC grant GP0036426, FCAR grant 89EQ4144 and partially by AFOSR grant 0066. We thank Nicole Paradis for her help in drawing the figures.     

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

Determination of radiative lifetimes in neutral nitrogen using short laser pulses from a distributed feedback dye laser

Radiative lifetimes were determined for two short-lived states in neutral nitrogen. Following photo-dissociation and two-photon excitation employing the same UV source, excitation to higher states was performed with a distributed feedback dye laser (DFDL). The lifetimes were found to be τ(2p ²4d ⁴ D _7/2)=17(3) ns and τ(2p ²5s ⁴ P _5/2)=22(3) ns.     

Spektrophotometrische Messung an kleinsten Flüssigkeitsmengen

Zusammenfassung Bei jeder spektrophotometrischen Meßeinrichtung ist das Volumen aller Küvetten, die den gleichen Lichtstrom hindurchtreten lassen, proportional dem Quadrat der Küvettenlänge. Der Proportionalitätsfaktor hängt bei Filterphotometern nur von den Eigenschaften von Lichtquelle und Empfänger ab, bei Verwendung eines Monochromators außerdem von den Abmessungen des Monochromators, im wesentlichen von seinem Auflösungsvermögen. Bei handelsüblichen Spektralphotometern ist es im Grünen möglich, bei 10 cm Schichtlänge mit einem Volumen von weniger als 0,1 ml ohne Einschränkung der Meßgenauigkeit auszukommen.

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Summary In every spectrophotometric measuring device the volume of all cells, which permit the passage of the same light flux, is proportional to the square of the length of the cell. In the case of filter photometers, the proportionality factor depends solely on the characteristics of the light source and receiver; if a monochromator is used also on the dimensions of the monochromator, and essentially on its resolving powers. With the spectrophotometers on the market, it is possible in the green, when using a 10 cm path length, to get along with a volume of less than 0.1 ml without decreasing the precision of measurement.

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Résumé Dans toutes les installations de spectrophotométrie le volume de toutes les cuves qui laissent passer la même quantité de lumière est proportionnel au carré de leur longueur. Le facteur de proportionnalité dépend dans les photomètres à filtre, des propriétés de la source de lumière et du récepteur. En outre lorsqu'on utilise un monochromateur il dépend des caractéristiques de ce dernier, c'est-à-dire essentiellement de son pouvoir séparateur. Dans les spectrophotomètres du commerce on peut opérer avec une couche de 10 cm de long et un volume inférieur à 0,1 ml sans diminuer la précision de la mesure.

     

Zur Luminiszenz von Cr(3)-Verbindungen

Whereas 2.4 is likely to be an upper limit for theDq/B-value of Cr(3)-complexes which show fluorescence the existence of a lower bound is rather unlikely. Experimental evidence suggests that the energy difference between the minima of the² E _g- and the⁴ T _2g-potential curves is one of the factors determining the relative intensities of phosphorescence and fluorescence.     

Electric field-controlled water permeation coupled to ion transport through a nanopore

We report molecular dynamics simulations of a generic hydrophobic nanopore connecting two reservoirs which are initially at different Na(+) concentrations, as in a biological cell. The nanopore is impermeable to water under equilibrium conditions, but the strong electric field caused by the ionic concentration gradient drives water molecules in. The density and structure of water in the pore are highly field dependent. In a typical simulation run, we observe a succession of cation passages through the pore, characterized by approximately bulk mobility. These ion passages reduce the electric field, until the pore empties of water and closes to further ion transport, thus providing a possible mechanism for biological ion channel gating.     

Thermische Reaktionen mit 3-Phenyl-2H-azirinen; 1, 3-dipolare Cycloadditionen und En-Reaktionen

Benzonitrile p-nitrobenzylide ( 5 ) undergoes 1,3-dipolar cyclo-additions in the presence of 3-phenyl-2H-azirines ( 1 ), yielding in benzene at 0° 2-(p-nitrophenyl)-4,5-diphenyl-1,3-diazabicyclo[3.1.0]hex-3-enes ( 7 , scheme 2). Under the basic conditions of the reaction mixture, 7 a and 7 b are partially converted to 2-(p-nitrophenyl)-4,5-diphenyl-1,6-dihydropyrimidines ( 8a, b ) which are dehydrogenated by oxygen to the corresponding pyrimidines 9a and 9b , respectively. 3-Phenyl-2H-azirines ( 1 ) form, on heating at 145° in xylene in the presence of the azalactone 32 (2,4-diphenyl-Δ²-oxazolin-5-one), 4-(aziridin-2′-yl)-2,4-diphenyl-Δ²-oxazolin-5-ones ( 33 , scheme 11). 33 arises from an ene reaction of the enol form of 32 with 1 . Similar ene reactions are observed with the azirines 1 and dimedone ( 37 , scheme 12). Under the ene reaction conditions (xylene, 145°), the non-isolated intermediate primary adducts ( 38a and 38b ) undergo rearrangements of the vinylcyclopane-cyclopentene type to give 6,6-dimethyl-4-oxo-1,3-diphenyl-4, 5, 6, 7-tetrahydroisoindole ( 40 ) and 6, 6-dimethyl-4-oxo-3-phenyl-4, 5, 6, 7-tetrahedroindole ( 42 ), respectively.     

A 2pπ, 3dπ valence bond wave function for the 1B1u state of ethylene

A 2p, 3d valence bond wave function for the lowest ¹ B _1u state of the -system of ethylene is variationally optimized with respect to the atomic orbital exponents using a non-empirical -approximation. The resulting energy compares favourably with previous calculations and leads to a satisfactory value for the lowest ¹ A _1g-¹ B _1u transition energy. The optimized exponent for the 2p orbital is close to the Slater value whereas the exponent for the 3d orbital is found to be nearly hydrogenic. The implications of this result are discussed in some detail.

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Zusammenfassung Für den tiefsten ¹ B _1u -Zustand des -Elektronensystems des Äthylens wird nach der Variations-methode eine 2p, 3d-Valenzbindungs-Wellenfunktion bezüglich der Orbitalkoeffizienten optimiert, wobei eine nicht-empirische -Approximation zugrunde gelegt wird. Die berechnete Energie stimmt gut mit den Ergebnissen vorausgegangener Rechnungen überein. Für den tiefsten ¹ A _1g ¹ B _1u -Übergang ergibt sich ein befriedigender Energiewert. Der optimierte Orbitalkoeffizient für das 2p-Orbital stimmt gut mit dem nach der Slaterschen Regel bestimmten Koeffizient überein, während man für den 3d-Orbitalkoeffizienten einen Wert findet, der dem des Wasserstoffs ähnlich ist.

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Résumé Une founction d'onde V.B.(2p , 3d ) est optimée pour l'état ¹ B _1u d'aethylen. L'energie calculé aussi que l'energie de transition la plus basse ¹ A _1g -¹ B _1u est en bon accord avec des valeurs connues.

     

Molekülstruktur in der Theorie der Spinvalenz I

Zusammenfassung Wie in der Theorie der Valenzstrukturen (Kimball 1940), so kann man auch in der Theorie der Spinvalenz ohne Rechnung, d. h. nur mit den Mitteln der Gruppen- und Darstellungstheorie, zu Aussagen über die Molekülstruktur von Verbindungen des Typs ML _n kommen. Die Ergebnisse für sog. ¹ A ₁-Moleküle werden angegeben und mit denen von Kimball verglichen.

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Spin valence theory provides a basis for the group theoretical derivation of the structure of ML _n -molecules. Results for molecules in a totally symmetrical ground state are compared with predictions of the valence bond method (Kimball 1940).

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Résumé La théorie de la valence de spin permet de dériver les structures de molécules du type ML _n á l'aide de la théorie des groupes. Des résultats pour de molécules en état ¹ A ₁ (totalement symétrique) sont comparés á ceux de la méthode de mésomérie (Kimball 1940).

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Auszugsweise vorgetragen auf der Westdeutschen Chemiedozententagung, Freiburg i. Brsg., 27.–30. 4. 1964 und der 8. I. C. C. C., Wien, 7.–11. 9. 1964.